A383880 a(n) = [x^n] 1/Product_{k=0..n-1} (1 - k*x)^2.
1, 0, 3, 72, 2307, 95060, 4817990, 290523576, 20333487251, 1621036680120, 145057745669850, 14399349523416000, 1570425994090538574, 186674663305762642296, 24021930409036829669036, 3327140929951823209016400, 493515678917684006649451651, 78054583374364036172432641200
Offset: 0
Keywords
Programs
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Mathematica
Join[{1}, Table[Sum[StirlingS2[n + k - 1, n - 1]*StirlingS2[2*n - k - 1, n - 1], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 14 2025 *) -
PARI
a(n) = polcoef(1/prod(k=0, n-1, 1-k*x+x*O(x^n))^2, n);
Formula
a(n) = Sum_{k=0..n} Stirling2(n+k-1,n-1) * Stirling2(2*n-k-1,n-1) for n > 0.
a(n) ~ 3^(3*n - 3/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * 2^(2*n - 1/2) * exp(n) * (3 - 2*w)^n * w^(2*n - 3/2)), where w = -LambertW(-3*exp(-3/2)/2). - Vaclav Kotesovec, May 14 2025